For each of the following samples that were given an experimental treatment, test whether these samples represent populations that are different from the general population: (a) a sample of 10 with a mean of 44, (b) a sample of 1 with a mean of 48. The general population of individuals has a mean of 40, a standard deviation of 6, and follows a normal curve. For each sample, carry out a Z test using the five steps hypothesis testing with a two-tailed test at the .05 significance level, and make a drawing of the distributions involved.
To test whether the samples represent populations that are different from the general population, we can perform a Z test using the five steps of hypothesis testing. Here are the steps:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis (H0): The mean of the sample is equal to the mean of the general population.
- Alternative hypothesis (Ha): The mean of the sample is different from the mean of the general population.
Step 2: Set the significance level (α):
In this case, the significance level is given as .05, which means we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Collect and summarize the data:
We are given two samples: Sample (a) with a sample size of 10 and a sample mean of 44, and Sample (b) with a sample size of 1 and a sample mean of 48. We also know that the general population has a mean of 40 and a standard deviation of 6, following a normal distribution.
Step 4: Calculate the test statistic:
For each sample, we can calculate the Z-test statistic using the formula:
Z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
For Sample (a):
Z(a) = (44 - 40) / (6 / sqrt(10))
For Sample (b):
Z(b) = (48 - 40) / (6 / sqrt(1))
Step 5: Determine the critical region and make a decision:
Since we are conducting a two-tailed test, we need to consider both tails of the distribution. The critical region is determined based on the significance level and the distribution (normal distribution in this case).
For a significance level of .05, the critical Z-value for a two-tailed test is approximately ±1.96.
If the absolute value of the calculated Z-test statistic falls outside the critical region (i.e., exceeds ±1.96), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
By making a drawing of the distributions involved, you can visually identify the critical region and determine whether the calculated Z-test statistic falls within or outside this region for each sample.
Note: The sketches/drawings involve plotting the normal distribution curves for the general population mean of 40 and standard deviation of 6, marking the critical region boundaries at ±1.96, and indicating the sample means on the x-axis.
Once you have calculated the Z-test statistics and made the drawings, compare the absolute values of the calculated Z-values with the critical value of ±1.96. If any of the calculated Z-values fall outside this range, you would reject the null hypothesis for that sample.